that vanishes for undeformed systemOne can very well work with the second Hamiltonian form for the KdV
system [1]}: \(H_2[u]=-\int _{-\infty }^\infty dx~\frac{1}{2}u^2(x)\) , with the alternate fundamental bracket defined as
\(\left\lbrace u(x),u(y)\right\rbrace =\left[\partial ^3_x+2\left(u_x+u\partial _x\right)\right]\delta (x-y)\) . Then, the time component
of the Lax pair will take the form: \(B\equiv -u_x\sigma _3-\left[u_{xx}-4\frac{\delta H_2[u]}{\delta x}\right]\sigma _++2u\sigma _-\) .
Rest will follow through the replacement:
\(\frac{2}{3}\left(\frac{\delta H_1[u]}{\delta u(x)}+u_{xx}\right)\rightarrow 4\frac{\delta H_2[u]}{\delta x}\) .. In
the presence of this anomaly, implementation of the deformed `equation of motion' (EOM) (i. e., the KdV equation),
\(&&u_t+u_{xxx}-\frac{2}{3}\partial _x\left(\frac{\delta H_1[u]}{\delta u(x)}+u_{xx}\right)+2uu_x=0,\nonumber \\{\rm or},\quad &&u_t+6uu_x+u_{xxx}={\cal X}_x,\) 
